23 April 2010
Archive for Rational Mechanics and Analysis
We prove that,if u:Ω ⊂ ℝn → ℝN is a solution to the Dirichlet variational problem involving a regular boundary datum (u0, ∂Ω) and a regular integrand F(x, w, Dw) strongly convex in Dw and satisfying suitable growth conditions, then Hn-1 -almost every boundary point is regular for u in the sense that Du is Hölder continuous in a relative neighborhood of the point. The existence of even one such regular boundary point was previously not known except for some very special cases treated by Jost & Meier (Math Ann 262:549-561, 1983). Our results are consequences of new up-to-the-boundary higher differentiability results that we establish for minima of the functionals in question. The methods also allow us to improve the known boundary regularity results for solutions to non-linear elliptic systems, and, in some cases, to improve the known interior singular sets estimates for minimizers. Moreover, our approach allows for a treatment of systems and functionals with "rough" coefficients belonging to suitable Sobolev spaces of fractional order. © 2010 Springer-Verlag.
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